Questions and Answers of First Course Differential Equations

A critical point c of an autonomous first-order DE is said to be isolated if there exists some open interval that contains c but no other critical point. Can there exist an autonomous DE of the
Consider the autonomous DE dy/dx = (2/πy)y - sin y. Determine the critical points of the equation. Discuss a way of obtaining a phase portrait of the equation. Classify the critical points as
Consider the autonomous differential equation dy/dx = f (y), where the graph of f is given. Use the graph to locate the critical points of each differential equation. Sketch a phase portrait of each
Find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand,
Consider the autonomous rst-order differential equation dy/dx = y2 - y4 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.(a) y0
Consider the autonomous first-order differential equation dy/dx = y - y3 and the initial condition y(0) = y0. By hand, sketch the graph of a typical solution y(x) when y0 has the given values.(a)
(a) Identify the nullclines (see Problem 17) in Problems€‘1, 3, and 4. With a colored pencil, circle any lineal elements in figures (1), (2) and (3) that you think may be a lineal element
For a first-order DE dy/dx = f (x, y) a curve in the plane defined by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use
(a) Consider the direction field of the differential equation dy/dx = x(y - 4)2 - 2, but do not use technology to obtain it. Describe the slopes of the lineal elements on the lines x = 0, y = 3, y
In parts (a) and (b) sketch isoclines f (x, y) = c for the given differential equation using the indicated values of c. Construct a direction field over a grid by carefully drawing lineal elements
The given figure represents the graph of f (y) and f (x), respectively. By hand, sketch a direction field over an appropriate grid for dy/dx = f (y) (Problem 13) and then for dy/dx = f (x)
Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points.1. y' = x(a) y(0) =
Reproduce the given computer-generated direction field. Then sketch, by hand, an approximate solution curve that passes through each of the indicated points. Use different colored pencils for

Showing 2500 - 2600 of 2513